15-Puzzle
State Space Search
Initial State
Goal State
1 2 3 4
5 6
8
9 10 7 12
13 14 11 15
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
NJIT CS288
1/10
Fall 2016 A.Sohn
Depth First Search
1 2 3 4
5 6
8
9 10 7 12
13 14 11 15
Up
1
Right
Down
Up
1 2 3 4
5 6 8
9 10 7 12
13 14 11 15
2
Down
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
Right
1 2 3 4
5 6 7 8
9 10 12
13 14 11 15
6
Right
Left
1 2
4
5 6 3 8
9 10 7 12
13 14 11 15
3
Up
4
1 2 3 4
5
6 8
9 10 7 12
13 14 11 15
5
8
1 2 3 4
5 6 7 8
9
10 12
13 14 11 15
9
11
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
12
Left
1 2 3 4
5 6
8
9 10 7 12
13 14 11 15
7
Up
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
NJIT CS288
Right
Down
Down
1 2 3 4
5 6 7 8
9 10 12
13 14 11 15
Left
Left
1 2 3 4
5 6 7 8
9 10 12
13 14 11 15
10
2/10
Fall 2016 A.Sohn
Depth First Search
iter
open before
0
1
1
1
2
2, 3, 4, 5
3
4
closed current
succ
closed
open after
1
1
2, 3, 4, 5
1
2, 3, 4, 5
1
2
6, 7, 8 , 9
1,2
6,7,9,3,4,5
6,7,9,3,4,5
1,2
6
10, 11 ,12
1,2,6
10,12,7,9,3,4,5
10,12,7,9,3,4,5
1,2,6
10
---
1,2,6,10
---
5
NJIT CS288
3/10
Fall 2016 A.Sohn
Breadth First Search
1 2 3 4
5 6
8
9 10 7 12
13 14 11 15
1
Up
Down Right
Up
Left
Left
1 2 3 4
5 6 7 8
9 10 12
13 14 11 15
1 2 3 4
5 6 8
9 10 7 12
13 14 11 15
1 2
4
5 6 3 8
9 10 7 12
13 14 11 15
1 2 3 4
5
6 8
9 10 7 12
13 14 11 15
2
3
4
5
Down
Right
Up
Left
Down
Right
Down
Right
Up
Left
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
1 2 3 4
5 6 7 8
9 10 12
13 14 11 15
1 2 3 4
5 6
8
9 10 7 12
13 14 11 15
1 2 3 4
5 6 7 8
9
10 12
13 14 11 15
1 2 3 4
5 6 8 12
9 10 7
13 14 11 15
1 2 3
5 6 8 4
9 10 7 12
13 14 11 15
1 2 3 4
5 6
8
9 10 7 12
13 14 11 15
6
7
8
9
10
11
12
NJIT CS288
4/10
Fall 2016 A.Sohn
Breadth First Search
iter
open before
0
1
1
1
2
2, 3, 4, 5
3
4
closed current
succ
closed
open after
1
1
2, 3, 4, 5
1
2, 3, 4, 5
1
2
6, 7, 8 , 9
1,2
3,4,5,6,7,9
3,4,5,6,7,9
1,2
3
10, 11, 12
1,2,3
4,5,6,7,9,11,12
4,5,6,7,9,11,12
1,2,3
4
---
1,2,3,4
---
5
NJIT CS288
5/10
Fall 2016 A.Sohn
Best First Search 1
1 2 3 4
5 6
8
9 10 7 12
13 14 11 15
1
Up
Down Right
Up
Left
Left
1 2 3 4
5 6 7 8
9 10 12
13 14 11 15
1 2 3 4
5 6 8
9 10 7 12
13 14 11 15
1 2
4
5 6 3 8
9 10 7 12
13 14 11 15
1 2 3 4
5
6 8
9 10 7 12
13 14 11 15
2
3
4
5
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
0
0
0
0
0
0
0
0
0
0
2
1
0
0
0
1
h=4
NJIT CS288
0
0
0
0
0
0
0
0
0
1
1
1
h=6
0
2
0
1
0
0
0
0
4
1
1
1
0
0
0
1
Down
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
goal state
0
0
0
0
Right
0
0
0
0
h=8
0
4
0
0
0
1
1
1
h=8
6/10
0
0
0
1
1. Compute heuristic values
2. Pick the best one
Fall 2016 A.Sohn
Best First Search 2
1 2 3 4
5 6
8
9 10 7 12
13 14 11 15
1
DN
1 2 3 4
5 6 7 8
9 10 12
13 14 11 15
RT
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
DN
1 2 3 4
5 6 8
9 10 7 12
13 14 11 15
2
4
DN
UP
RT
RT
1 2
4
5 6 3 8
9 10 7 12
13 14 11 15
1 2 3 4
5 6
8
9 10 7 12
13 14 11 15
4
8
1 2 3 4
5
6 8
9 10 7 12
13 14 11 15
5
8
8
6
1 2 3 4
5 6 7 8
9
10 12
13 14 11 15
9
6
11
4
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
12
5
LT
7
4
UP
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
NJIT CS288
3
6
UP
1 2 3 4
5 6 7 8
9 10 12
13 14 11 15
6
2
LT
LT
1 2 3 4
5 6 7 8
9 10 12
13 14 11 15
10
0
7/10
Fall 2016 A.Sohn
Best First Search
iter open before closed current
0
1
1
1
2
2, 3, 4, 5
3
6,7,9,3,4,5
NJIT CS288
succ
closed
open after
1
1
2(h=4), 3(h=6),
4(h=8), 5(h=8)
1
2(h=4), 3(h=6),
4(h=8), 5(h=8)
1
2
6(2), 7(4), 8 , 9(6)
1,2
6(2), 7(4), 9(6),
3(6), 4(8), 5(8)
1,2
6
10(0),11(4)
,12(5)
1,2,6
10(0), 7(4),
12(5),9(6), 3(6),
4(8), 5(8)
8/10
Fall 2016 A.Sohn
A* Search
1 2 3 4
5 6
8
9 10 7 12
13 14 11 15
1
DN
RT
1 2 3 4
5 6 7 8
9 10 12
13 14 11 15
2
5=1+4
DN
RT
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
6
4=2+2
DN
RT
UP
1 2 3 4
5 6 8
9 10 7 12
13 14 11 15
UP
1 2 3 4
5 6 7 8
9 10 12
13 14 11 15
UP
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
NJIT CS288
LT
3
7=1+6
1 2
4
5 6 3 8
9 10 7 12
13 14 11 15
1 2 3 4
5 6
8
9 10 7 12
13 14 11 15
4
9=1+8
1 2 3 4
5
6 8
9 10 7 12
13 14 11 15
5
9=1+8
8
8=2+6
1 2 3 4
5 6 7 8
9
10 12
13 14 11 15
9
8=2+6
11
7=3+4
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
12
8=3+5
LT
7
6=2+4
LT
10
3=3+0
9/10
1 2 3 4
5 6 7 8
9 10 12
13 14 11 15
Fall 2016 A.Sohn
A* Search
iter open before closed current
0
1
1
1
2
2, 3, 4, 5
3
7, 3, 9, 4, 5
NJIT CS288
succ
closed
open after
1
1
2(f=g+h=5),
3(f=g+h=7),
4(f=g+h=9),
5(f=g+h=9)
1
2(f=5), 3(f=7),
4(f=9), 5(f=9)
1
2
6(4), 7(6), 8(8) ,
9(8)
1,2
6(4), 7(6), 3(7),
9(8), 4(9), 5(9)
1, 2
6
10(3),11(7)
,12(8)
1,2,6
10(3), 7(6), 3(7),
12(8), 9(8), 4(9),
5(9)
10/10
Fall 2016 A.Sohn